A081567 -id:A081567 - cp's OEIS frontend

A147748 Row sums of Riordan array ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2)).

1, 2, 6, 20, 70, 250, 900, 3250, 11750, 42500, 153750, 556250, 2012500, 7281250, 26343750, 95312500, 344843750, 1247656250, 4514062500, 16332031250, 59089843750, 213789062500, 773496093750, 2798535156250, 10125195312500

Offset: 0

Paul Barry, Nov 11 2008

Row sums of A147747. Binomial transform of A061646.
Counts all paths of length (2*n), n>=0, starting at the initial node on the path graph P_9, see the Maple program. - Johannes W. Meijer, May 29 2010
From L. Edson Jeffery, Apr 19 2011: (Start)
For the 5 X 5 unit-primitive matrix (see [Jeffery])
A_(10,1) = [0,1,0,0,0; 1,0,1,0,0; 0,1,0,1,0; 0,0,1,0,1; 0,0,0,2,0],
a(n) = (Trace([A_(10,1)]^(2*n)))/5. (See also A189315.) (End)

S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
L. Edson Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A000984, A030191, A061646, A081567, A147746, A147747, A178381, A189315.

with(GraphTheory): G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=24; n2:=nmax*2: for n from 0 to n2 do B(n):=A^n; a(n):= add(B(n)[1,k], k=1..9); od: seq(a(2*n), n=0..nmax); # Johannes W. Meijer, May 29 2010

(1 - 3x + x^2)/(1 - 5x + 5x^2) + O[x]^25 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 05 2016 *)

G.f.: (1-3*x+x^2)/(1-5*x+5*x^2).
a(n) = 5*a(n-1) - 5*a(n-2) for n > 2, a(0)=1, a(1)=2, a(2)=6. - Philippe Deléham, Nov 13 2008
For n >= 1: a(n) = (2/5)*((5-sqrt(5))/2)^n + (2/5)*((5+sqrt(5))/2)^n. - Richard Choulet, Nov 14 2008
G.f.: 1/(1-2x/(1-x/(1-x/(1-x)))) (hence sequence approximates A000984 in first few terms). - Paul Barry, Aug 05 2009
a(n) = (1/5)*Sum_{k=1..5} (x_k)^(2*n), x_k=2*cos((2*k-1)*Pi/10). - L. Edson Jeffery, Apr 19 2011
From R. J. Mathar, Apr 20 2011: (Start)
a(n) = A030191(n) - 3*A030191(n-1) + A030191(n-2).
a(n) = 2*A081567(n-1), n > 0. (End)
a(n) = Sum_{k=0..n} A147746(n,k)*2^k. - Philippe Deléham, Oct 30 2011
E.g.f.: (1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2))/5. - Stefano Spezia, Jul 09 2024

A217770 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=4 or if k-n >= 6, T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 0, 1, 5, 10, 10, 4, 0, 0, 6, 15, 20, 14, 0, 0, 0, 6, 21, 35, 34, 14, 0, 0, 0, 0, 27, 56, 69, 48, 0, 0, 0, 0, 0, 27, 83, 125, 117, 48, 0, 0, 0, 0, 0, 0, 110, 208, 242, 165, 0, 0, 0, 0, 0, 0, 0, 110, 318, 450, 407, 165

Offset: 0

Philippe Deléham, Mar 24 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
n=0: 1, 1,  1,  1,   1,   1,   0,   0,    0,    0,    0, 0, ...
n=1: 1, 2,  3,  4,   5,   6,   6,   0,    0,    0,    0, 0, ...
n=2: 1, 3,  6, 10,  15,  21,  27,  27,    0,    0,    0, 0, ...
n=3: 1, 4, 10, 20,  35,  56,  83, 110,  110,    0,    0, 0, ...
n=4: 0, 4, 14, 34,  69, 125, 208, 318,  428,  428,    0, 0, ...
n=5: 0, 0, 14, 48, 117, 242, 450, 768, 1196, 1624, 1624, 0, ...
...
Square array, read by rows, with 0 omitted:
...1,    1,     1,     1,     1,      1
...1,    2,     3,     4,     5,      6,      6
...1,    3,     6,    10,    15,     21,     27,     27
...1,    4,    10,    20,    35,     56,     83,    110,    110
...4,   14,    34,    69,   125,    208,    318,    428,    428
..14,   48,   117,   242,   450,    768,   1196,   1624,   1624
..48,  165,   407,   857,  1625,   2821,   4445,   6069,   6069
.165,  572,  1429,  3054,  5875,  10320,  16389,  22458,  22458
.572, 2001,  5055, 10930, 21250,  37639,  60097,  82555,  82555
2001, 7056, 17986, 39236, 76875, 136972, 219527, 302082, 302082
...
Triangle begins:
1
1, 1
1, 2,  1
1, 3,  3,  1
1, 4,  6,  4,  0
1, 5, 10, 10,  4,  0
0, 6, 15, 20, 14,  0, 0
0, 6, 21, 35, 34, 14, 0, 0
...

Cf. Similar sequences: A214846, A216054, A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235, A216236, A216238, A217257, A217315, A217593, A217765.

T(n,n+4) = T(n,n+5) = A094788(n+2).
T(n,n+3) = A217783(n).
T(n,n+2) = A217779(n).
T(n,n+1) = A081567(n).
T(n,n)   = A217782(n).
T(n+1,n) = A217778(n).
T(n+3,n) = T(n+2,n) = A094667(n+1).
Sum(T(n-k,k), k=0..n) = A217777(n).

A270863 Self-composition of the Fibonacci sequence.

Original entry on oeis.org

0, 1, 2, 6, 17, 50, 147, 434, 1282, 3789, 11200, 33109, 97878, 289354, 855413, 2528850, 7476023, 22101326, 65338038, 193158521, 571033600, 1688143881, 4990651642, 14753839486, 43616704857, 128943855250, 381196100507, 1126928202714, 3331532438042, 9848993360069

Offset: 0

Oboifeng Dira, Mar 24 2016

This sequence has the same relation to the Fibonacci numbers A000045 as A030267 has to the natural numbers A000027.
From Oboifeng Dira, Jun 28 2020: (Start)
This sequence can be generated from a family of composition pairs of generating functions g(f(x)), where k is an integer and where
f(x) = x/(1-k*x-x^2) and g(x) = (x+(k-1)*x^2)/(1-(3-2*k)*x-(3*k-k^2-1)*x^2).
Some cases of k values are:
k=-5, f(x) g.f. 0,A052918(-1)^n and g(x) g.f. 0,A081571
k=-4, f(x) g.f. A001076(-1)^(n+1) and g(x) g.f. 0,A081570
k=-3, f(x) g.f. A006190(-1)^(n+1) and g(x) g.f. 0,A081569
k=-2, f(x) g.f. A215936(n+2) and g(x) g.f. 0,A081568
k=-1, f(x) g.f. A039834(n+2) and g(x) g.f. 0,A081567
k=0,  f(x) g.f. A000035 and g(x) g.f. 0,A001519(n+1)
k=1,  f(x) g.f. A000045 and g(x) g.f. A000045
k=2,  f(x) g.f. A000129 and g(x) g.f. 0,A039834(n+1)
k=3,  f(x) g.f. A006190 and g(x) g.f. 0,A001519(-1)^n
k=4,  f(x) g.f. A001076 and g(x) g.f. 0,A093129(-1)^n
k=5,  f(x) g.f. 0,A052918 and g(x) g.f. 0,A192240(-1)^n
k=6,  f(x) g.f. A005668 and g(x)=(x+5*x^2)/(1+9*x+19*x^2)
k=7,  f(x) g.f. 0,A054413 and g(x)=(x+6*x^2)/(1+11*x+29*x^2).
(End)

			a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50.

Colin Barker, Table of n, a(n) for n = 0..1000
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Oboifeng Dira, Family of composition pairs g(f(x)) generating A270683
Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).

Cf. A000027, A000045, A001622, A030267, A000035, A001519, A000129, A039834.

I:=[0, 1, 2, 6]; [m le 4 select I[m] else 3*Self(m-1)+Self(m-2)-3*Self(m-3)-Self(m-4): m in [1..30]]; // Marius A. Burtea, Aug 03 2019

f:= x-> x/(1-x-x^2):
a:= n-> coeff(series(f(f(x)), x, n+1), x, n):
seq(a(n), n=0..30);

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-3,1,3]^(n-1)*[1;2;6;17])[1,1] \\ Charles R Greathouse IV, Mar 24 2016

concat(0, Vec(x*(1-x-x^2)/(1-3*x-x^2+3*x^3+x^4) + O(x^40))) \\ Colin Barker, Mar 24 2016

a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=2, a(3)=6.
G.f.: x*(1-x-x^2) / (1-3*x-x^2+3*x^3+x^4). - Colin Barker, Mar 24 2016
G.f.: B(B(x)) where B(x) is the g.f. of A000045. - Joerg Arndt, Mar 25 2016
a(n) = (phi*((phi^2 + 5^(1/4)*sqrt(3*phi))^n - (phi^2 - 5^(1/4)*sqrt(3*phi))^n) + (psi^2 + 5^(1/4)*sqrt(3*psi))^n - (psi^2 - 5^(1/4)*sqrt(3*psi))^n)/(2^n * 5^(3/4) * sqrt(3*phi)), where phi = (sqrt(5) + 1)/2 is the golden ratio, and psi = 1/phi = (sqrt(5) - 1)/2. - Vladimir Reshetnikov, Aug 01 2019
0 = a(n)*(a(n) +6*a(n+1) -a(n+2)) +a(n+1)*(8*a(n+1) -9*a(n+2) +a(n+3)) +a(n+2)*(-8*a(n+2) +6*a(n+3)) +a(n+3)*(-a(n+3)) if n>=0. - Michael Somos, Feb 05 2022

A206947 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank above 0.

Original entry on oeis.org

0, 0, 0, 2, 14, 70, 306, 1248, 4888, 18666, 70110, 260414, 959882, 3519232, 12854064, 46824210, 170243566, 618125238, 2242100898, 8126927456, 29442587720, 106626616954, 386046638142, 1397431266222, 5057790129274, 18304064121600, 66237312391776

Offset: 0

David Nacin, Feb 13 2012

Here, the term uniform used in the sense of Retakh, Serconek and Wilson. Graded is used in terms of Stanley's definition that all maximal chains have the same length n.

R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A206948 (removing unique maximal element.)

Cf. A206949, A206950 (allowing one or two elements in each rank level above 0 with and without maximal element.)

Join[{0}, LinearRecurrence[{8, -21, 20, -5}, {0, 0, 2, 14}, 40]]

def a(n,adict={0:0,1:0,2:0,3:2,4:14}):
    if n in adict:
        return adict[n]
    adict[n]=8*a(n-1)-21*a(n-2)+20*a(n-3)-5*a(n-4)
    return adict[n]

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(1)=0, a(2)=0, a(3)=2, a(4)=14.
G.f.: (2*(1-x)*x^3)/((1-3*x+x^2)*(1-5*x+5*x^2)).
a(n) = A081567(n-1) - A001906(n).

A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.

Original entry on oeis.org

1, 0, 3, 0, 1, 7, 0, 1, 3, 17, 0, 1, 3, 10, 41, 0, 1, 3, 11, 30, 99, 0, 1, 3, 12, 35, 87, 239, 0, 1, 3, 13, 40, 108, 245, 577, 0, 1, 3, 14, 45, 130, 322, 676, 1393, 0, 1, 3, 15, 50, 153, 406, 938, 1836, 3363, 0, 1, 3, 16, 55, 177, 497, 1236, 2682, 4925, 8119, 0, 1

Offset: 1

Clark Kimberling, Feb 25 2012

Row sums, u(n,1): (1,2,5,13,...), odd-indexed Fibonacci numbers.
Row sums, v(n,1): (1,3,8,21,...), even-indexed Fibonacci numbers.
As triangle T(n,k) with 0<=k<=n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

			First five rows:
  1
  0   3
  0   1   7
  0   1   3   17
  0   1   3   10   41
First five polynomials u(n,x):
  1, 3*x, x + 7*x^2, x + 3*x^2 + 17*x^3, x + 3*x^2 + 10*x^3 + 41*x^4.

Cf. A208344.

Cf. A084938, A000045, A000244, A001906.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208344 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208345 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]
Table[v[n, x] /. x -> 1, {n, 1, z}]

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 26 2012: (Start)
As triangle T(n,k), 0<=k<=n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-2) - 2*T(n-2,k-1) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1+(y-1)*x)/(1-(1+2*y)*x+y*(2-y)*x^2).
Sum_{k=0..n} T(n,k)*x^k = A152167(n), A000007(n), A001906(n+1), A003948(n) for x = -1, 0, 1, 2 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A078057(n), A001906(n+1), A000244(n), A081567(n), A083878(n), A165310(n) for x = 0, 1, 2, 3, 4, 5 respectively. (End)

A328646 Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (1-x)/(x^2-3x+1)).

Original entry on oeis.org

1, -1, 2, -2, 1, 5, -6, 3, -1, 13, -20, 12, -4, 1, 34, -65, 50, -20, 5, -1, 89, -204, 195, -100, 30, -6, 1, 233, -623, 714, -455, 175, -42, 7, -1, 610, -1864, 2492, -1904, 910, -280, 56, -8, 1, 1597, -5490, 8388, -7476, 4284, -1638, 420, -72, 9, -1, 4181

Offset: 0

Clark Kimberling, Nov 01 2019

The first 501 polynomials are irreducible. Column 1 of the array: A001519 (odd-indexed Fibonacci numbers). Row sums: A000045 (Fibonacci numbers). Alternating row sums: essentially 5*A081567.

			First eight rows:
    1,    -1;
    2,    -2,    1;
    5,    -6,    3,    -1;
   13,   -20,   12,    -4,   1;
   34,   -65,   50,   -20,   5,   -1;
   89,  -204,  195,  -100,  30,   -6,   1;
  233,  -623,  714,  -455, 175,  -42,   7, -1;
  610, -1864, 2492, -1904, 910, -280,  56, -8,  1;
First eight polynomials:
1 - x
2 - 2 x + x^2
5 - 6 x + 3 x^2 - x^3
13 - 20 x + 12 x^2 - 4 x^3 + x^4
34 - 65 x + 50 x^2 - 20 x^3 + 5 x^4 - x^5
89 - 204 x + 195 x^2 - 100 x^3 + 30 x^4 - 6 x^5 + x^6
233 - 623 x + 714 x^2 - 455 x^3 + 175 x^4 - 42 x^5 + 7 x^6 - x^7
610 - 1864 x + 2492 x^2 - 1904 x^3 + 910 x^4 - 280 x^5 + 56 x^6 - 8 x^7 + x^8

Cf. A328647, A001519, A000045, A081567.

g[x_, n_] := Numerator[ Factor[D[(1 - x)/(x^2 - 3 x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x]
Table[h[n], {n, 0, 10}]
Column[%] (* A328646 array *)

A122068 Expansion of x(1-3x)(1-x)/(1-7x+14x^2-7x^3).

Original entry on oeis.org

1, 3, 10, 35, 126, 462, 1715, 6419, 24157, 91238, 345401, 1309574, 4970070, 18874261, 71705865, 272491891, 1035680954, 3936821259, 14965658694, 56893879910, 216295686467, 822315097387, 3126323230541, 11885921055638

Offset: 1

Gary W. Adamson, Oct 15 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
R. Witula, P. Lorenc, M. Rozanski, and M. Szweda, Sums of the rational powers of roots of cubic polynomials, Zeszyty Naukowe Politechniki Slaskiej, Seria: Matematyka Stosowana z. 4, Nr. kol. 1920, 2014.
Index entries for linear recurrences with constant coefficients, signature (7,-14,7).

Cf. A087946, A081567.

Cf. A215007, A215008. - Roman Witula, May 16 2014

a:=[1,3,10];; for n in [4..30] do a[n]:=7*(a[n-1]-2*a[n-2]+a[n-3]); od; a; # G. C. Greubel, Oct 03 2019

I:=[1,3,10]; [n le 3 select I[n] else 7*(Self(n-1) -2*Self(n-2) + Self(n-3)): n in [1..30]]; // G. C. Greubel, Oct 03 2019

seq(coeff(series(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), x, n+1), x, n), n =1..30); # G. C. Greubel, Oct 03 2019

M = {{2,1,0,0,0,0}, {1,2,1,0,0,0}, {0,1,2,1,0,0}, {0,0,1,2,1,0}, {0,0,0, 1,2,1}, {0,0,0,0,1,2}}; v[1] = {1,1,1,1,1,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n,30}]
Rest@CoefficientList[Series[x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), {x, 0, 30}], x] (* G. C. Greubel, Oct 03 2019 *)
LinearRecurrence[{7,-14,7},{1,3,10},30] (* Harvey P. Dale, Mar 08 2020 *)

Vec(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)+O(x^30)) \\ Charles R Greathouse IV, Sep 27 2012

def A122068_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)).list()
a=A122068_list(30); a[1:] # G. C. Greubel, Oct 03 2019

From Roman Witula, May 16 2014: (Start)
a(n) = (1/2)*Sum_{k=0..2}(1 - 1/sqrt(7)*cot(2^k * alpha))* (2*sin(2^k * alpha))^(2n), where alpha := 2*Pi/7.
a(n) = (A215007(n) + A215008(n+1) - 2*A215008(n))/2. (End)
a(n) = binomial(2*n-1, n-1) + Sum_{k=1..n} (-1)^k*binomial(2*n, n+7*k). - Greg Dresden, Jan 28 2023

A217778 Expansion of (1-x)^2(1-3x)/((1-3x+x^2)(1-5x+5x^2)).

Original entry on oeis.org

1, 3, 10, 34, 117, 407, 1429, 5055, 17986, 64278, 230473, 828391, 2982825, 10754459, 38811802, 140165322, 506449789, 1830590295, 6618524221, 23933966743, 86562282258, 313102489406, 1132598701585, 4097213146599, 14822370816337, 53623952036787

Offset: 0

Philippe Deléham, Mar 24 2013

A diagonal of the square array A217770.

Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A217770.

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)^2*(1-3*x)/((1-3*x+x^2)*(1-5*x+5*x^2)))); // Bruno Berselli, Mar 28 2013

LinearRecurrence[{8, -21, 20, -5}, {1, 3, 10, 34}, 26] (* Bruno Berselli, Mar 28 2013 *)
CoefficientList[Series[(1-x)^2(1-3x)/((1-3x+x^2)(1-5x+5x^2)),{x,0,30}],x] (* Harvey P. Dale, Sep 26 2023 *)

makelist(expand(((3+sqrt(5))*(5+sqrt(5))^n-(3-sqrt(5))*(5-sqrt(5))^n+(1+sqrt(5))*(3+sqrt(5))^n-(1-sqrt(5))*(3-sqrt(5))^n)/(4*2^n*sqrt(5))), n, 0, 25); /* Bruno Berselli, Mar 28 2013 */

G.f.: (1-5*x+7*x^2-3*x^3)/(1-8*x+21*x^2-20*x^3+5*x^4).
a(n) = A081567(n) - A094865(n).
a(n) = A217770(n+1,n).
a(n) = 8*a(n-1) -21*a(n-2) +20*a(n-3) -5*a(n-4) for n>3, a(0)=1, a(1)=3, a(2)=10, a(3)=34.
a(n) = ((3+r)*(5+r)^n-(3-r)*(5-r)^n+(1+r)*(3+r)^n-(1-r)*(3-r)^n)/(4*r*2^n), where r=sqrt(5). [Bruno Berselli, Mar 28 2013]

A112091 Number of idempotent order-preserving partial transformations (of an n-element chain).

Original entry on oeis.org

1, 2, 6, 21, 76, 276, 1001, 3626, 13126, 47501, 171876, 621876, 2250001, 8140626, 29453126, 106562501, 385546876, 1394921876, 5046875001, 18259765626, 66064453126, 239023437501, 864794921876, 3128857421876, 11320312500001

Offset: 0

Abdullahi Umar, Aug 25 2008

			a(2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 2-element chain), namely: the empty map, (1)->(1), (2)->(2), (1,2)->(1,1), (1,2)->(1,2), (1,2)->(2,2); the mappings are coordinate-wise.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Callan and T. Mansour, Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns, arXiv:1705.00933 [math.CO] (2017), Table 2 No 200 (offset 1 then).
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8
Index entries for linear recurrences with constant coefficients, signature (6,-10,5).

[ n eq 1 select 1 else n eq 2 select 2 else n eq 3 select 6 else 6*Self(n-1)-10*Self(n-2)+ 5*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 21 2011

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==1+5(a[n-1]-a[n-2])},a[n], {n,30}] (* or *) LinearRecurrence[{6,-10,5},{1,2,6},31] (* Harvey P. Dale, Aug 20 2011 *)

Vec((2*x-1)^2/(1-x)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = ((sqrt(5))^(n - 1))*(((sqrt(5) + 1)/2)^n - ((sqrt(5) - 1)/2)^n) + 1. [corrected by Jason Yuen, Sep 06 2024]
a(n) = 1 + 5*(a(n-1) - a(n-2)), a(0) = 1, a(1) = 2.
G.f.: (1 - 2*x)^2/((1 - x)*(1 - 5*x + 5*x^2)). Convolution of A081567 with the sequence 1, -1, -1, -1 (-1 continued). - R. J. Mathar, Sep 06 2008
a(n) = 1 + A030191(n-1). - R. J. Mathar, Jun 20 2011
a(n) = 6*a(n-1) - 10*a(n-2) + 5*a(n-3); a(0) = 1, a(1) = 2, a(2) = 6. - Harvey P. Dale, Aug 20 2011
E.g.f.: exp(x) + (exp((5 + sqrt(5))*x/2) - exp((5 - sqrt(5))*x/2))/sqrt(5). - Franck Maminirina Ramaharo, Nov 09 2018

A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 18, 8, 1, 16, 56, 41, 11, 1, 32, 160, 170, 73, 14, 1, 64, 432, 620, 377, 114, 17, 1, 128, 1120, 2072, 1666, 704, 164, 20, 1, 256, 2816, 6496, 6608, 3649, 1178, 223, 23, 1, 512, 6912, 19392, 24192, 16722, 7001, 1826, 291, 26, 1, 1024, 16640, 55680, 83232, 69876, 36365, 12235, 2675, 368, 29, 1

Offset: 0

Paul Barry, Nov 15 2005

Row sums are A081567. Diagonal sums are A085810. Product of Pascal triangle A007318 and Morgan-Voyce triangle A085478.

Unsigned version of A123876. - Philippe Deléham, Oct 25 2007

			Triangle begins:
   1;
   2,   1;
   4,   5,   1;
   8,  18,   8,  1;
  16,  56,  41, 11,  1;
  32, 160, 170, 73, 14, 1;
  ...

Cf. A081567, A085810, A007318, A085478, A123876.

T(2n,n) gives A026000.

Number triangle T(n,k) = Sum_{j=0..n} C(n, j)*C(j+k, 2k);
T(n,k) = Sum_{j=0..n} C(n, k+j)*C(k, k-j)*2^(n-k-j);
T(n,k) = Sum_{j=0..n-k} C(n+k-j, n-k-j)*C(k, j)*(-1)^j*2^(n-k-j).
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 4*T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014

More terms from Michel Marcus, Sep 09 2024

cp's OEIS Frontend

A147748 Row sums of Riordan array ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2)).

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A217770 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=4 or if k-n >= 6, T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A270863 Self-composition of the Fibonacci sequence.

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A206947 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank above 0.

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A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.

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A328646 Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (1-x)/(x^2-3x+1)).

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A122068 Expansion of x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3).

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A217778 Expansion of (1-x)^2*(1-3*x)/((1-3*x+x^2)*(1-5*x+5*x^2)).

A122068 Expansion of x(1-3x)(1-x)/(1-7x+14x^2-7x^3).

A217778 Expansion of (1-x)^2(1-3x)/((1-3x+x^2)(1-5x+5x^2)).